Find the standard error of a sample mean and the probability of observing a given sample mean from a population.
Result
How to use this tool
Use the known or assumed population mean μ.
Use σ, or a sample estimate if σ is unknown.
Larger samples give smaller standard errors.
Enter the observed mean to get its z-score and probability.
Formula & worked example
With population mean 100, standard deviation 15 and samples of 25, the standard error is 3, and a sample mean of 103 sits at z = 1 (about 84.13% of sample means fall below it).
Limitations — read this before you rely on it
Probabilities use the normal approximation. For small samples from non-normal populations, or with estimated σ, the t-distribution is more accurate.
Frequently asked questions
The distribution of a statistic, here the sample mean, across all possible samples of the same size.
Larger samples average out individual variation, so sample means cluster tighter around the population mean.
The central limit theorem says sample means are approximately normal for large n (often n ≥ 30 is used as a rule of thumb).
σ describes individuals; the standard error describes sample means and is always smaller for n > 1.
As an estimate, yes, but strictly the t-distribution then applies, especially for small samples.